Characterization of the turbulent transport in the edge plasma of the tokamak ISTTOK

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UNIVERSIDADE DE LISBOA

FACULDADE DE CIÊNCIAS

DEPARTAMENTO DE FÍSICA

Characterization of the turbulent transport in the edge plasma

of the tokamak ISTTOK

Sara Vaz Mendes

Mestrado Integrado em Engenharia Física

Dissertação orientada por:

Carlos Alberto Nogueira Garcia da Silva

Olinda Maria Quelhas Fernandes Conde

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Acknowledgments

This thesis presents an effort as a master student to participate in the large effort of the fusion community in trying to understand the major challenges of terrestrial fusion. I am really grateful for the opportunity of finishing my masters in this subject of studies. More than anything, I would like to thank the availability and kindness of my supervisor, Dr. Carlos Silva. The opportunity to analyze and perform a characterization of the turbulence in the edge of the ISTTOK tokamak, along with all the knowledge about fusion devices that I got to take a first look at has been more than I could have hoped for the end of my master program. Thank you to Dr. Carlos Silva, here as a professor, for helping me in spite of my almost a year-long absence through Erasmus. Thanks to the IPFN team at IST for creating the space and means for an academic Fusion program in Lisbon.

A very special gratitude goes out to all responsible for my student internships at ENEA and CCFE for the great experiences to further my knowledge in Fusion diagnostics. These opportunities were only possible because of the great research conditions and hospitality at both institutions. Thanks to professor Guiomar Evans for the availability and guidance through our most pessimist times at the faculty. Thanks to both professors Guiomar Evans and Olinda Conde for the help in solving the not so straight forward administration procedures and allowing me to follow the last master subjects in this area.

Lastly, I also have to thank all my colleagues and family for the last years hearing my pessimist self, while still encouraging me about all the work ahead. Thank you all for the patient

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Abstract

A major challenge in the realization of fusion power plants will be to overcome the dominant confinement losses induced by turbulence. The performance of current experiments is strongly influenced by these mechanisms. Turbulence considerably increases the transport of particles and energy in the edge, being the dominant source of losses in fusion devices. This work aims at contributing to the understanding of the turbulence induced by blob-filaments, trough a multi-scale investigation of the induced fluctuations on plasma quantities and ultimately the resultant transport of particles and energy in the edge of theISTTOK(ISTtokamak) experiment. It was intended to carry a detailed analysis of turbulence physics at various scales, from their origin to the impact on plasma confinement. ISTTOK is an ideal experiment to carry a study related to the edge plasma since it is compact, flexible and allows rapid installation of diagnostics. A multi-pin array of Langmuir probes allowed to carry bi-dimensional measurements of the structure of plasma fluctuations, and revealed to be ideal in determining the induced particle transport.

The different analysis techniques applied in this work revealed that the turbulent structures in the edge of the ISTTOK tokamak have a time scale of the order of ∼ 2 − 10 µs, propagating poloidally with vθ = 5 − 10 km/s and having a poloidal dimension of ∼ 2 − 6 cm.

Fluctua-tion levels from 25% in the edge plasma to 150% in the far scrape-off layer (SOL) of ISTTOK revealed that the region is dominated by fluctuations, which are probably induced by the tur-bulence resulting in blob-filaments. The remaining statistical properties of the fluctuations also indicated that the edge andSOLofISTTOKare dominated by fluctuations, where intermittency in plasma quantities results in probability distributions with high skewness and kurtosis. The statistical quantities of the fluctuations allowed to show the increase of the relative importance of the induced fluctuations with radius.

Further investigations were done on the properties of the edge/SOL fluctuations reveling broad power spectra, and coherent with structures of the order of 10 − 100 µs. The power

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spectra of the fluctuations when appropriately re-scaled to the standard deviation of the mea-surements shows a roughly universal shape in most fusion devices. The common result was also seen for the power spectra computed during this work, showing a modest decay with frequency until ∼ 100 kHz, and a faster decay from this value with the power law of f−C.

Lastly, the particle flux induced by the fluctuations in the edge and SOL was estimated to be in the order of ∼ 1021 m−2s−1, and having the same order magnitude of the total particle losses at ISTTOKestimated from the measured particle confinement.

Keywords

Thermonuclear Fusion; Tokamak; Turbulence structures; Blob-filaments; Intermittent particle transport.

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Resumo

Um enorme esfor¸co por parte da comunidade de fusão nuclear tem vindo a reunir nas últimas décadas vários estudos sobre as principais dificuldades em construir futuros reatores baseados na fusão nuclear. Na actualidade, várias experiências espalhadas pelo globo têm como finalidade provar a eficiência de diferentes configura¸cões para futuros reatores alimentados por rea¸cões de fusão nuclear. Uma das configura¸cões mais promissoras para o sucesso da tecnologia de fusão nuclear na Terra é o Tokamak. Um dos principais obstáculos desta experiência deve-se a mecan-ismos de turbulência na periferia do plasma. De modo geral a eficiência das atuais experiências de fusão é fortemente influenciada por estes mecanismos de turbulência. A turbulência aumenta consideravelmente o transporte de part´ıculas e energia no plasma periférico, sendo a fonte dom-inante de perdas em experiências de fusão nuclear. A turbulência verificada na região periférica resulta em filamentos de plasma (regiões de densidade superior). Estes filamentos nascem e propagam-se no plasma periférico. Um resultado fundamental da propaga¸cão de filamentos de plasma são as flutua¸cões induzidas em parâmetros do plasma, tal como a densidade e o potencial.

Neste trabalho pretendeu-se compreender a f´ısica dos filamentos de plasma resultantes da turbulência, através de uma investiga¸cão das flutua¸cões induzidas nos parâmetros do plasma e, finalmente, o transporte de part´ıculas e energia resultante na perifer´ıa do ISTTOK (IST Tokamak). Executou-se uma análise detalhada da f´ısica da turbulência a várias escalas, desde a sua origem até ao impacto no confinamento do plasma. O ISTTOKé uma experiência ideal para realizar estudos relacionados com o plasma periférico, uma vez que é compacto, flex´ıvel e permite a instala¸cão rápida de diagnósticos [1,2]. Um extenso programa de Fusão Nuclear tem vindo a ser implementado no IST (Instituto Superior Técnico) com diversos diagnósticos instalados, entre os quais as sondas de Langmuir. As sondas de Langmuir consistem basicamente em eletrodos cil´ındricos que podem ser inseridos no plasma periférico (até alguns cent´ımetros dentro da última superf´ıcie de fluxo fechada). As sondas de Langmuir utilizadas no diagnóstico do plasma doISTTOKtêm uma elevada resolu¸cão espacial (de alguns mil´ımetros) e temporal (na ordem de microsegundos). Adicionalmente, noISTTOKdiversos estudos foram implementados

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ao longo dos anos com sistemas de múltiplas sondas de Langmuir [2–5]. Durante este trabalho foi usado um sistema de várias sondas separadas poloidalmente, permitindo determinar a estrutura poloidal das fluctua¸cões. Este diagnóstico mostrou-se ideal na determina¸cão do transporte de part´ıculas induzido pelas estruturas de turbulência em propaga¸cão no plasma.

As sondas de Langmuir permitem tra¸car uma curva caracter´ıstica I-V (onde V é a diferen¸ca de potencial aplicada à sonda e I a corrente recolhida por esta). Em determinadas condi¸cões, aplicando uma alta diferen¸ca de potencial negativa, a corrente recolhida corresponde a uma corrente de satura¸cão iónica I_sat+ . Por outro lado, pode não ser aplicada qualquer diferen¸ca de potencial à sonda de modo a medir o potencial flutuante Vf. As flutua¸cões de Isat+ permitem

estimar as flutua¸c˜oes da densidade de plasma, enquanto as flutua¸c˜oes de Vf permitem estimar

as flutua¸c˜oes do potencial de plasma.

Através da aplica¸cão de diferentes técnicas de análise, desde análise estat´ıstica até técnicas de correla¸cão e análise espectral, as flutua¸cões foram caracterizadas, contribuindo assim para uma melhor compreensão da f´ısica associada aos filamentos do plasma e o seu impacto no trans-porte radial.

OISTTOKpode operar com ciclos de corrente alternada, para os quais a dire¸cão da corrente de plasma é invertida periodicamente, permitindo obter descargas mais longas. Com a inversão entre ciclos positivos e negativos (dire¸cão da corrente de plasma) surgem também altera¸cões na posi¸cão do plasma. Durante este trabalho a caracteriza¸cão das flutua¸cões e do transporte in-duzido noISTTOKteve especial aten¸cão a este facto. Foi analisada a importância das estruturas de turbulência ao longo de seis ciclos consecutivos de corrente alternada no ISTTOK.

As diversas propriedades estat´ısticas das flutua¸cões aqui investigas permitiram concluir que o plasma periférico e aSOLdoISTTOK são dominados por flutua¸cões. Os n´ıveis de flutua¸cões elevados, na orderm de 25% numa região dentro do limitador para 150% naSOLdoISTTOK, foi o primeiro resultado apresentado a comprovar que a região é dominada por flutua¸cões, que são provavelmente induzidas pela turbulência. De seguida, os elevados valores de skewness e kurtosis da densidade e do potencial de plasma foram apresentados, de modo a indicar mais uma vez a importância das flutua¸cões no plasma periférico do ISTTOK. Estas quantidades estat´ısticas permitiram mostrar o aumento da importância relativa das flutua¸cões induzidas com o aumento do raio.

No cap´ıtulo 2 foram mencionados estudos anteriores sobre a importância das flutua¸cões no plasma periférico. De um dos exemplos foi real¸cada a “universalidade” da Fun¸cão Densidade de Probabilidade (FDP) para as flutua¸cões do plasma periférico. Este resultado foi também conclu´ıdo para as flutua¸cões no plasma do ISTTOK, no cap´ıtulo4.

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Adicionalmente, o espectro de frequências das flutua¸cões foi investigado. A análise espectral dos sinais de Vf e de Isat+ obtidos no ISTTOK revelaram que a potência espectral é

domi-nada por frequências na gama 10-100 kHz, e que a amplitude diminui com a frequência de acordo com 1/fC, onde C é uma constante que varia nas diferentes regiões do espectro. De modo geral o espectrograma das flutua¸cões do plasma periférico é largo em frequência. Os es-pectros de frequência tornam-se mais largos com o aumento do raio, indicando o aumento da importância relativa das altas frequências sobre as baixas frequências. Os espectrogramas obti-dos assemelham-se ao espectro de pink noise que é bastante usual em sinais intermitentes, como o que se dá para os parâmetros do plasma devido a propaga¸cão das estruturas de turbulência.

O sistema de sondas utilizado permite determinar Vf e Isat+ para diferentes posi¸c˜oes poloidais

noISTTOK. No total temos sete sondas espa¸cadas entre si por 2 mm. Os sinais obtidos para as diferentes posi¸cões foram comparados através da fun¸cão correla¸cão cruzada, de modo a deter-minar a velocidade de propaga¸cão, e as estruturas espacial e temporal das flutua¸cões na dire¸cão poloidal. A correla¸cão entre as medi¸cões nas diferentes posi¸cões poloidais é dependente da di-mensão poloidal das estruturas intermitentes, assim como do tempo caracter´ıstico e velocidade de propaga¸cão destas através do plasma. Os resultados para a correla¸cão cruzada entre dois sinais de Vf ou de Isat+ indicam a similaridade entre os sinais em fun¸cão do desfasamento

tem-poral entre os dois. As diferentes técnicas de análise aplicadas durante este trabalho revelaram que as estruturas de turbulência na periferia do tokamakISTTOKtêm uma estrutura temporal na ordem de ∼ 2 − 10 µs, em propaga¸cão na dire¸cão poloidal com vθ= 5 − 10 km/s e dimensão

poloidal na ordem de ∼ 2 − 6 cm.

O fluxo médio de part´ıculas que é induzido pelas flutua¸cões pode ser estimado através das medi¸cões simultâneas das flutua¸cões de densidade e de potencial. Tivemos durante este trabalho a oportunidade de enfatizar a grande influência deste fluxo induzido nas perdas totais no plasma periférico de um tokamak. O fluxo de part´ıculas induzido pelas flutua¸cões na periferia do ISTTOK encontra-se na ordem de 1021 _m−2_s−1_{, e apresenta a mesma ordem de grandeza das}

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perdas totais de part´ıculas no ISTTOK estimadas a partir do confinamento de part´ıculas medido.

Palavras Chave

Fus˜ao Termonuclear; Tokamak; Estruturas de turbulˆencia; Filamentos; Transporte intermitente de part´ıculas.

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List of Figures

1.1 General scheme of a typical tokamak. [Adapted figure from [6]]. . . 5 1.2 General scheme of the toroidal and poloidal directions and fields in a tokamak

device. Main geometric parameters of the device. [Adapted figure from [7]]. . . . 6 1.3 General schemes for the cross section of a tokamak with a divertor or with a

magnetic separatrix. Common scenarios for the plasma-material interface. For devices with a divertor the plasma-material interaction takes place further away from the confined plasma, in a localized region on the vessel’s edge (the diver-tor target plates). Extra coils are responsible for producing the X-point in the poloidal magnetic field. Near the X-point are the divertor target plates, set to better withstand plasma interactions. [Adapted figure from [8]]. . . 9 1.4 Lorentz force and Larmor motion. Uniform circular motion in the plane

perpen-dicular to the magnetic field, with rL and wL. Helical path for charged particles with a velocity parallel to the magnetic field. [Adapted figure from [9]]. . . 10 2.1 Results from cameras installed in the JET tokamak. In the left is showing a

2D density plot (intensity map) in the perpendicular plane. In the perpendicu-lar view, high density blob-like structures are shown just outside the separatrix (dashed black line). On the right, the high density structures at the JET tokamak are seen as brighter filaments, extending along the toroidal field (dashed red line). [Adapted figure from [10]]. . . 19 2.2 (a, b) Camera Image from the MAST tokamk. Images obtained from the MAST

tokamak (a,b) are processed in order to evidence the high density filament struc-tures corresponding to brighter pixels (c). The filament strucstruc-tures not only show to extended onward in the chamber, but also to be highly aligned with the toroidal magnetic field, having the same helical geometry along the torus. [This figure is a cortesy of Tom Farley (MAST, CCFE), 2019].. . . 21

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2.3 Gas-puff imaging (GPI) frames from the NSTX tokamak, taken near the outer midplane separatrix (solid line). The frames, with a frame rate of 7.5 µs/frame and a ∼ 25 × 25 cm2 field of view, show the formation of a density blob (brighter feature) near the separatrix and its subsequent propagation radially outwards with ∼ Vr= 1 km/s. A poloidal motion is also observable, although more subtle. [Figure from [11]]. . . 22 2.4 2D density plots from the edge of the DIII-D tokamak, obtained with the beam

emission spectroscopy (BES) diagnostic. The Fig. shows two frames, where the frame rate is 6 µs, and each frame covers a 6 × 5 cm2 area at the edge plasma of the DIII-D tokamak. While red indicates high density features, blue indicates low density ones. In the frames a high density feature of spatial structure of ∼ 2 × 2 cm2, marked with a dashed circle, shows to propagate both poloidally and radially over the 2D perpendicular plane, with ∼ Vr = 1.5 Km/s and ∼ Vθ = 5 Km/s. [Figure from [12,13]].. . . 22 2.5 Scheme of the convective radial drift of a blob filament, resulting from the charge

polarization mechanism. A ~E × ~B velocity component (~V_{E× ~}~ _B ≡ ~V_E~ ≡ ~Vr) dom-inates the propagation of plasma filaments. The dominant radial component is induced by a charge polarizing force ( ~F ) in the same direction. [Illustration from [14]]. . . 23 2.6 Langmuir Probe measurements obtained from the DIII-D tokamak. 1 ms time

recordings of the I_sat+ (t) ( ˜I_sat+ (t) ∝ ˜n(t)), Vf(t) ( ˜Vf(t) ∝ ˜Vp(t)), and poloidal electric field Eθ. The results display frequent burst-like events above the signals rms. Finally it is also shown a sample of the product IsatEθ which is scalable with the intermittent radial particle flux (~neV~r= nθ/Bφ). [Figure from [12]]. . . 25 2.7 Conditional averaging results for Langmuir Probe measurements obtained from

the DIII-D tokamak. The intermittent bursts on measurements taken at different radius in the scrape-off layer plasma are shown. Within (a) 0.5 cm, (b) 5 cm and (c) 10 cm of the LCFS at r = Rsep. Averaging over 20-40 events in each signal was taken for events over a 2.5 rms-level threshold (set initially to discriminate the bursts in the ion saturation current fluctuations). [Figure from [12]]. . . 26 2.8 Semilogarithmic plot of the PDFs of the ion saturation current, normalized to the

standard deviation. Results determined on the Tore Supra (solid line), Alcator C-Mod (thick solid line), MAST (dashed–dotted line), and PISCES (dots). [Figure from [15]]. . . 28

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2.9 Power Spectra of the density intermittency obtained in the edge of various stel-larator and tokamak devices. [Figure from [16]]. Results taken from Langmuir Probe measurements at the radius where the poloidal turbulence flow speed was near zero, [16]. In order to reveal that all curves have nearly the same shape, the factors presented in the legend were used to re-scaled the frequencies, and the amplitudes normalized. . . 29 3.1 The ISTTOK tokamak experiment at IST. A small circular cross-section device,

with a large aspect ratio, having minor radius a = 8.5 cm and major radius R = 46 cm, and a graphite limiter set at rL ' 8.5 cm. An extensive scientific programme is carried at the the ISTTOK tokamak with the various diagnostics seen in the image. . . 32 3.2 Scheme of the main diagnostics installed at ISTTOK, along with their toroidal

positions. The ports used in the diagnosis of the plasma are also shown. . . 33 3.3 Poloidal rail limiter installed at ISTTOK (at rL' 8.5 cm). A graphite material

with the shape of the vacuum vessel, that extends along the poloidal perimeter with an interrupted structure. The smaller diameter of the limiter will interrupt the magnetic field lines trajectory, in an attempt to prevent the conduction of plasma particles towards the walls. . . 34 3.4 Shot lists for the initial study of the intermittent fluctuations carried at the

ISTTOK tokamak. Analysis for two different plasma current values, IpA = 4

kA (shot list A) and IpB = 4.7 kA (shot list B). Measurements were taken at

different radial positions across the ISTTOK boundary plasma, from r = 70 mm to r = 105 mm. . . 37 3.5 Shot lists for a second study of the intermittent fluctuations carried at the ISTTOK

tokamak (shot list C and shot list D). Analysis for different pressure values and gas injection regimes. Measurements were taken at different radial positions across the ISTTOK boundary plasma, from r = 80 mm to r = 95 mm, with Bφ' 0.5 T IpC,D ∼ 4 kA. . . 38

3.6 Time recordings of the density, plasma current and Vloop signals during a full discharge on tokamak ISTTOK (shot 41169, at r = 90 mm). A total of six alternating current cycles allowed to obtain coherent discharges (of ∼ 160 ms), with individual cycles having flat top regimes of ∼ 20 ms. . . 39

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3.7 Time recordings of the plasma position during a full discharge on tokamak ISTTOK (for the discharge shown in Fig. 3.6). A total of six alternating current cycles allowed to obtain coherent discharges (of ∼ 160 ms), with individual cycles having flat top regimes of ∼ 20 ms. . . 40 3.8 Radial profile for the electronic temperature, Te, on the ISTTOK tokamak.

Pre-vious measurements for established radial positions (black triangles), along with an exponential interpolation for the remaining edge/SOL locations in study on chapter 4. . . 41 3.9 On the left its shown an illustrative scheme of the diagnosis of the plasma with

Langmuir probes. Usually a cylindrical pin electrode that is polarized by an external circuit, and then inserted in the plasma. On the right, presented the characteristic I-V curve between the collected current by the probe, Ipr, and the potential applied to it, Vpr. From which it is possible to determine local plasma quantities, such as the electron temperature, Te, electron density, ne ' n, and plasma potential, Vp. . . 43 3.10 (a) Scheme of the insertion of the poloidal array of Langmuir probes on ISTTOK.

Trough a port that allows to diagnose the edge plasma with the possibility to radially shift the probe systems from shot to shot by means of a drive mechanism. (b) Photograph of the probe system. Consisting of a 7-pin poloidal array of probes, sequentially separate by 2 mm. Probes with 0.75 mm of diameter and 2 mm of length. . . 44 3.11 Scheme of the poloidal probe array operations mode. The channels from table

3.2 are also identified. . . 46

3.12 Example of the determination of the I_sat+ fluctuations. Moving Average smoothing result for a standard I_sat+ signal from ISTTOK with N=1000. The smoothed trend for a time scale of ∼ 1 ms is shown in red. [Figure from [17]]. . . 47

3.13 Time delay analysis. Example scheme of the correlation for two signals taken in poloidally shifted positions. [Adapted figure from [17]]. . . 49

3.14 Scheme a probe system configuration used to previously measure Vf signals. Cor-relation between the signals acquired by the 7 poloidally shifted probes. . . 50

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4.1 Langmuir Probe time recordings of the ion saturation current and floating poten-tial, at the edge of the ISTTOK tokamak (r = 80 mm), for #41174. The time series were obtained with the sampling rate of 2 MHz. It is shown the overall intermittent character of the edge fluctuations in the ISTTOK tokamak, and an enlargement on the burst-events temporal structures. . . 53

4.2 Radial profile of the ion saturation current (mean values), taken with Langmuir probe recordings at the edge and SOL of the ISTTOK tokamak. The radial measurements represent the mean values for probe samples of about ∼ 3 − 5 ms, with the sampling rate of 2 MHz. The results from shot list A (black squares) were taken for Ip = 4 kA, while the results from shot list B (red triangles) with Ip = 4.7 kA, Fig. 3.4. The remaining device/plasma parameters were kept for all discharges.. . . 54

4.3 Radial profile of the floating potential (mean values), taken with Langmuir probe recordings at the edge and SOL of the ISTTOK tokamak. The radial profiles are computed in the same way as proceeded for the plots in Fig. 4.2. The results from shot list A (black squares) were taken for Ip = 4 kA, while the results from shot list B (red triangles) with Ip = 4.7 kA, Fig. 3.4. . . 55

4.4 Radial profile of plasma potential. Computed from the measurements of the floating potential profile on Fig. 4.3 along with the temperature profile for typical ISTTOK discharges, on Fig. 3.8, according to 3.2.. The results from shot list A (black squares) were taken for Ip = 4 kA, while the results from shot list B (red triangles) with Ip= 4.7 kA, Fig. 3.4. . . 56

4.5 Radial profile of the ion saturation current (mean values) for shot lists C and D. The radial profiles are computed in the same way explained for Fig.4.2. The results from shot list C (black squares and blue circles) were taken for p ∼ 1.5e−4 torr, while the results from shot list D (red triangles) with p ∼ 6.0e−4 torr, Fig. 3.5. . . 58

4.6 Radial profile of the floating potential (mean values) for shot lists C and D. The radial profiles are computed in the same way explained for Fig.4.2. The results from shot list C (black squares and blue circles) were taken for p ∼ 1.5e−4 torr, while the results from shot list D (red triangles) with p ∼ 6.0e−4 torr, Fig. 3.5. . 59

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4.7 Radial profile of the standard deviation (σ_I+

sat) and fluctuation level (σI + sat/ ¯I

+ sat) of the fluctuating ion saturation current. The results from shot list A (black squares) were taken for Ip = 4 kA, while the results from shot list B (red triangles) with Ip = 4.7 kA, Fig. 3.4.. . . 61 4.8 Radial profile of the skewness and kurtosis of the ion saturation current. The

results from shot list A (black squares) were taken for Ip = 4 kA, while the results from shot list B (red triangles) with Ip= 4.7 kA, Fig. 3.4. . . 62 4.9 Radial profile of the skewness and kurtosis of the floating potential. The results

from shot list A (black squares) were taken for Ip = 4 kA, while the results from shot list B (red triangles) with Ip = 4.7 kA, Fig. 3.4. . . 63 4.10 PDF of the ion saturation current time series. The distributions were computed

for Langmuir probe recordings of about ∼ 3 − 5 ms. The results presented were taken for the discharges #41174, #41172, #41169 and #41167, with Ip = 4 kA from list A, and #41219, #41216, #41214, and #41212 with Ip = 4.7 kA from list B, Fig.3.4. . . 64 4.11 PDF of the floating potential time series. The distributions were computed for

Langmuir probe recordings of about ∼ 3 − 5 ms. The results presented were taken for the discharges #41174, #41172, #41169 and #41167, with Ip = 4 kA from list A, and #41219, #41216, #41214 and #41212 with Ip = 4.7 kA from list B, Fig.3.4. . . 65 4.12 power spectra (PS) of the ion saturation current (solid lines) and floating potential

(dashed lines) time series at the edge/SOL of the ISTTOK tokamak. For each quantity was computed the PS for Ip = 4 kA (#41174, #41172, #41169 and #41167 from list A, Fig.3.4 ). The frequencies were normalized according to the factor 1/σ of each time series. The PSs’ shape supports what was described in chapter 2 (see Fig.2.9). . . 66 4.13 PDF and PS of the I_sat+ fluctuations. Comparison of the PDF and power spectra

results shown on Fig.4.10, Fig.4.11 and Fig.4.12. The fluctuations PDF shows a higher asymmetry (higher skewness) for the outwards locations. Most clrear for r=95 mm. While for the PS, the relative importance of the lower frequencies decreases with radius and broader spectras are observed. . . 67

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4.14 PDF and PS of the Vf fluctuations. Comparison of the PDF and power spectra results shown on Fig.4.10, Fig.4.11 and Fig.4.12. The relative importance of the lower frequencies decreases with radius and broader spectras are observed. Its known from Fig. 4.9 that the asymmetry of the PDF of ˜Vf increases with radius, although the changes are not very perceptible in this figure. . . 68 4.15 Smoothing result of a Savitzky-Golay filter (n=1001, M=4) on the initial

record-ings of the ion saturation current and floating potential (r = 80 mm, f = 2 MHz), for #41174. The technique ended up revealing in this case a possible trend for the structures first introduced in the zoom-in in Fig. 4.1 (at r=80 mm). . . 69 4.16 Radial profile of the blobs’ poloidal velocity, k~vθk, determined according to (4.7),

where it is used the cross correlations of both ion saturation current and floating potential signals. The blob-filaments propagate with opposing direction across the shear layer (∼ rL), however in both sides of the velocity shear layer are seen similar magnitudes. . . 71 4.17 Time delay analysis. The auto-correlation time for the I_sat+ measurements from

ch 11 were used to estimate the characteristic time structures of the fluctuations. 73 4.18 Characteristic spatial structures of the fluctuations.. . . 74 4.19 Radial profile of the turbulent particle flux, ¯Γr =< ˜n(t) ·E~˜θ(t) > /B. The plasma

density is estimated from fluctuating ion saturation current (channel 11), and the poloidal electric field fluctuations, E~˜θ, estimated from two floating potential signals ∆θ= 2 mm apart, (measurements from channels 13 and 14). . . 75

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List of Tables

3.1 ISTTOK geometric and usual discharge parameters. . . 35 3.2 Names of the diagnostics channels used during this thesis to access data from the

SDAS server. Its is also listed the quantities measured using this channels.. . . . 36 4.1 Determination of the decay with f−C for the I_sat+ spectra on Fig. 4.12 (at r=80

mm, during cycle 1). The constant C has been seen to take different values in different segments of the curve. . . 67

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Acronyms

ADC Analog-to-digital converter

Asdex Axially Symmetric Divertor Experiment

ATCA Advanced Telecommunications Computing Architecture BES Beam emission spectroscopy

CODAS Control and Data Acquisition System DAC Digital-to-analog converter

DAQ Data Acquisition

GPI Gas-puff imaging

IST Instituto Superior T´ecnico ISTTOK IST Tokamak

ITER International Thermonuclear Experimental Reactor JET Joint European Torus

LCFS Last closed flux surface

LP Langmuir Probe

MAST Mega Ampere Spherical Tokamak

MHD Magnetohydrodynamics

PDF Probability distribution function PFCs Plasma facing components

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RAM Random-Access Memory SDAS Shared DataAccess System SOL Scrape-off layer

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Chapter 1 Thermonuclear fusion

1.1 Introduction

To start the current work we present a simple fusion reaction

2

1H +31H →42 He(3.5M eV ) +10n(14.1M eV ) (1.1)

, that could be the solution to the energetic needs of the world. However, how simple it may look, to efficiently harness the energy of this reaction in a continuous and stable mode of operation on earth is not that easy. As it seems to always be fifty years way from taking place. The reaction presented on (1.1) for deuterium (2₁H) and tritium (3₁H) corresponds to a light ele-ment fusion reaction, in which these nuclei fuse to generate heavier α particles (4₂He), relishing a great amount of energy. In general a nuclear reaction (fusion and fission) releases about a factor of one million more energy when compared to the energy released in a chemical reaction, such as the ones involved in the combustion of fossil fuels.

Fossil fuels have been the main driver for the civilization’s growth and development into a more scientific and technological era. However, with the increase use of fossil fuels came an increase in the emission of green house gases well above the natural production range, busting from the Industrial Revolution, which was marked by a large combustion of coal. Coal has been the prevailing source of electricity and overall energy supply, not only when compared to the remaining fossil fuels, but to all energy sources. It was responsible for 39% of the world’s total electricity production in 2014, as shown for the most recent inquires of [18]. Furthermore, in 2014, the remaining electricity production was achieved 22% by gas, 17% by hydroelectric generation, 11% through nuclear sources, 5% through oil, and finally renewable energy sources

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entered with various others in a participation of only 7%, [18].

The repercussions on the environment that follow from carbon dioxide and other green house emissions have resulted in the increase of the average global temperatures. Followed by this came the raise in sea levels with growing rates, and on other climate changes that have imposed a large conscience about the use of fossil fuels. Although society has understood the repercus-sions of fossil fuel usage, major obstacles are still present for the remaining energy options. For example, renewable sources overall present efficiency instabilities based on the oscillations of the meteorologic conditions. Also require a large area to increase the environmental condi-tion’s exploitation. On the other hand, present nuclear fission power plants not only require high investments for construction, but are also involved in political conflicts. For these reasons fossil fuels will remain the main source of electricity production in the next years to come. In particular coal, whose world’s reserve could be used to produce electricity for another hundreds of years, given current world rate of consumption. Nevertheless, even before the limits of fossil fuel reserves are reached, given the environmental repercussions mentioned above, this source of energy production will have to be eventually reduced. Since the current options to fossil fuels don’t present to be efficient alternatives for the long run, a new scientific effort must be done to keep up with the seemingly never-ending growing needs of society. Not just in an economically positive way, but also presenting to be a less harmful alternative to the environment.

Thermonuclear fusion presents to be a very attractive option for producing uninterrupted electricity, with even higher gain in comparison to fission technology. In fission the bombardment of 235₉₂ U with a neutron results in 0.88 MeV per nucleon ( [19]), macroscopically equivalent to 84 × 106 MJ/kg. During the fusion of D-T nuclei the energy relished per reaction corresponds to 3.52 MeV per nucleon, which is macroscopically equivalent to 338 × 106 MJ/kg ( [19]). While deuterium nuclei are naturally abundant in earth’s oceans, there is no natural tritium on earth. It’s possible, however, to breed tritium using lithium. Very attractive and simple numbers presented by Dr Ian Chapman from the Culham Center for Fusion energy at the Royal Institution illustrated how one mole of the D-T nuclei could be used to produce about 1012 _{J of}

energy. The D-T nuclei are relatively abundant, and one mole (equivalent to a few grams of the reactants) found for example in a bathtub of water along with the lithium found in two laptop batteries. This rather accessible amount of reactants would nearly sustain the energy of one person for his entire adult life, assuming already a very high consumption lifestyle. The 17.6 MeV from the resulting products in (1.1), considering now 1 mol (6.03 × 1023particles), would result in about 1.65 × 1012J, and as suggested by Dr Ian Chapman to a 60 years energy supply

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of one person, assuming a 20 KWh/day consumption rate.

The understanding of the current limitations in achieving uninterrupted reactions in present thermonuclear devices is fundamental. It should provide the start for a path in the construction of economically feasible reactors with high power gain. Although a safe guess is that the costs of a fusion facility should be higher than current alternatives given the complex, and some, still undeveloped technology needed. However, still laying in a competitive range. Projects involving experimental machines designed to explore fusion, such as ISTTOK (the fusion experiment at Instituto Superior T´ecnico) and, in preparation, the big collaboration inITER(which will count with 35 nations to build the world’s largest tokamak in southern France) will contribute in this understanding. Making possible in the future the incorporation of the improved technology in complete electrical power plants (maybe just another fifty years).

Along with the great benefits attractive to fusion comes tremendous challenges to the physics and engineering communities, from plasma physics, transport theories, to material and electrical engineering. Nuclear fusion is an never-ending challenge provider.

In order for the positively charged nuclei to overcome the Coulomb repulsion among them and fuse, they will have to possess extremely high kinetic energies. The reaction presented on (1.1) is one of a couple of options for a device based on fusion reactions between light elements, and should be the main focus for fusion power plants. Mainly given to its higher cross-section at lower temperatures. In particular, temperatures of about 20 keV (over 200 million Kelvin (k)) are necessary for the reactants on (1.1) to fuse in a continuous and self-sustained state of ig-nition. Starting with an initial fuel of atomic deuterium and tritium, in the necessary kinetic energy conditions for the positive nuclei of the two species to come close and undergo a fusion reaction, the fuel will have then become a plasma, presenting outstanding temperatures way above the core of stars.

1.2 Thermonuclear Ignition

Great efforts have been done to construct a device capable of reaching stable thermonuclear ignition. Ignition implies maintaining a significant amount of plasma confined at the high temperatures needed, and during a sufficient time to result in a positive power balance. When these features have been achieved a fusion device could operate uninterruptedly being fed at a given rate with fuel. The Lawson criterion [19] and later the ”triple product”, given as

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nT τE ≥ fD−T ∼ (3 − 5) × 1021sm−3keV, ”triple product”

(1.2)

for the D-T reaction, express the conditions on plasma density (ne,i∼ n), temperature (Te,i∼ T )

and energy confinement time (τe) in order for a magnetic fusion device to reach thermonuclear

ignition. For a plasma in ignition (satisfying the criteria) the alpha power balances the energy sinks in the plasma, such as the irreducible Bremsstrahlung losses, [19]. Therefore, the fusion reactions in the plasma are self-sustained by one of the products they originate. These highly energetic alpha particles allow to maintain the plasma’s energy requirements through multiple collisions with the D and T nuclei.

For the D-T reaction the minimum of the triplet criterion occurs for temperatures around the T = 20 keV reference. Commonly, to reach ignition, a combination of ohmic and external auxiliary heating are used in the initial transient phase until a transition temperature. Previous to the transition temperature, the alpha heating is negligible against the losses due to thermal conduction and Bremsstrahlung radiation. Above it the alpha heating power becomes more significant and eventually rises the temperature to satisfy the”triple product”, according to the value of the two remaining quantities. Nonetheless, various combinations of the three parameters in the ”triple product” are considered among the fusion community to reach plasma ignition. Current devices have not yet been able to sustain simultaneously the parameters in order to reach and maintain a steady and continuous mode of operation with an ignited plasma. However, ITERis expected to exceed all current results.

1.3 Tokamak Principles

There are different experimental configurations still in consideration to reach on earth the plasma confinement requirements that allow to generate thermonuclear fusion power.

Far in the run are magnetic confinement devices, such as tokamaks (experiments as ISTTOK, ITER, JET,Asdex Upgrade, DIII-D, Tore Supra and Alcator C-Mod), stellarators (TJ-II and Wendelstein 7-X, for example) and also spherical torus (asMAST). The main principle of these devices is a system of magnetic fields responsible for the confinement of the plasma charged particles in a vacuum chamber.

The charged alpha particles that result from (1.1) are confined in the chamber by the mag-netic fields, and their power should sustain the ignited plasma. On the other hand, the neutrons

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Figure 1.1 General scheme of a typical tokamak. [Adapted figure from [6]].

escape the magnetic fields and the plasma providing the main source of heat in material struc-tures (the blanket), and are ultimately responsible for the production of electricity as commonly seen for other electricity sources, via restive heating, water vessels and turbines.

The tokamak is the most promising fusion configuration. Leading the run to understand and achieve fusion power in a controlled manner, and also in one that will allow to obtain a consid-erable positive gain in the approaching ITER experiment. In a tokamak the plasma is kept in a vacuum chamber with the shape of a torus (toroidal magnetic chamber), as in the scheme on Fig. 1.1. The acronym comes from the Russian designation Toroidalnaya Kamera Magnitnaya. The device was first introduced in 1950’s by the work of Soviet physicists, [20].

The charged particles are kept in toroidal motion due to the Lorentz force. That causes the particles to orbit (with Larmor radius) along the toroidal magnetic field lines. The choice of a plasma vessel shaped as a torus is in order to avoid particle losses at the end of the fields lines. The radius of the center of the torus R = R0 is referred to as the major radius, and the radius

of the torus cross section r = a as the minor radius. Finally, R/a is referred to as the aspect ratio R0/a.

The toroidal field Bφ is generated by the coils that contour the chamber’s cross section, the

toroidal magnetic coils, in Figs. 1.1and 1.2. Nevertheless, the particles must spin along helical field lines, to balance overall velocity drift components that arise from the toroidal geometry of

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Figure 1.2 General scheme of the toroidal and poloidal directions and fields in a tokamak device. Main geometric parameters of the device. [Adapted figure from [7]].

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the magnetic field and therefore avoid an unstable assemble.

The geometry of the toroidal field results in vertical velocity drifts (sec. 1.4), namely the ∇ ~B and curvature drifts. That arise due to the field’s dependence on the major radius (| ~Bφ| ∝ 1/R)

and due to the curvature of the field lines, respectively. The components of these vertical drifts in opposite directions for ions and electrons establish a vertical electrical field that ultimately results in the ~E × ~B drift. The ~E × ~B is radially outwards for both ions and electrons, and therefore causes a coherent displacement of the plasma outwards into the vessel’s walls.

This setback for an initial toroidal field scheme (also known as the pure toroidal θ-pinch configuration) provides therefore an unstable confinement. To balance these vertical drifts of ions and electrons the field lines must wrap around the torus, as shown in Fig. 1.2. The particles in turn will still possess the vertical ∇ ~B and curvature drifts. However, as they move along helical field lines for a significant number of toroidal turns, the vertical ∇ ~B and curvature drifts are averaged out.

The helical magnetic field is attained (in a configuration known as the screw pinch) by the combination of the toroidal magnetic field Bφ with a small poloidal magnetic field Bθ. The

Bθ field results from a toroidal current circulating on the plasma (usually Bθ ∼ Bφ/10). This

current corresponds to the secondary winding of a transformer, which has a solenoid in the center of the torus as primary winding, seen in Fig. 1.1.

The safety factor, q, primarily represents the number of toroidal turns required for any given field line to perform one full poloidal turn. In the picture of macroscopic stability based on the MHD(Magnetohydrodynamics) model q is given by 1.3,

q(r) = rBφ(r) R0Bθ(r)

. (1.3)

This estimation holds for most axisymmetric toroidal configurations (and considering a cylin-drical, large aspect ratio approximation), [19]. Configurations with q > 2 at the edge tend to be more stable.

The toroidal, poloidal and radial directions for a tokamak plasma can be better understood in Fig. 1.2, respectively, along φ, θ and ~r. It is often referred to the direction along the magnetic field lines as simply the parallel direction, and perpendicularly to the magnetic field lines as just the perpendicular direction.

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which is established by a system of coils often located outside the first wall. These coils are referred to as the outer poloidal coils on Fig. 1.1, (in some schemes, can be also found a set of inner poloidal coils).

1.3.1 Scrape-off layer

The magnetic field lines and the current lines lie on surfaces of constant pressure. A set of nested toroidal surfaces in a well-confined MHD plasma equilibrium, [19]. Both the magnetic field and current lines have no component perpendicular to these nested surfaces, thus they are also commonly referred to as flux surfaces. The flux surfaces should be closed surfaces in the vacuum chamber to avoid particle loss. Nonetheless, in the exterior region they are inter-rupted by components of the chamber’s walls. The outer region of the chamber characteristic of having open magnetic surfaces (opened magnetic field lines) is usually referred to as scrape-off layer (SOL). The screw pinch configuration, used to confine particles along the helical field lines, should limit particle loss to result only from particle collisions along the radial direction. However, as desired to discuss in this work a solid understanding through the last decades has revealed that more complex plasma mechanisms such as high density plasma features (known as filaments or blob-filaments) appear and propagate in the SOLregion of fusion plasmas. It is experimentally observed that plasma losses are up to 100 times larger than the expected to oc-cur by particle collisions, most probably due to the filamentary structures. In fact these plasma blobs are also strongly influenced by the ∇ ~B and curvature drifts. Polarized blob structures occur in the plasma outer regions with the electric field on these structures providing a mecha-nism for their convective ~E × ~B drift.

The division between the core plasma (confined inside the closed flux surfaces) an the SOL is marked by the last closed flux surface (LCFS) in limited machines, or by the magnetic separa-trix in diverted machines. On Fig. 1.3are presented general schemes of theLCFSor separatrix considering the two common scenarios for the plasma-wall interface (limiter or divertor). A limiter is simply a material structure interrupting the field lines before the actual chamber walls, and therefore determining the LCFS. Limiters can be installed in various geometries. Commonly seen are poloidal limiters, which can be a material with the shape of the vacuum vessel, but having a smaller diameter, and extending completely along the poloidal perimeter. Could also have an interrupted structure (rail limiters). Limiters can even be assembled as just a localized solid component in the vessel. Moreover a material interface placed in a certain local

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Figure 1.3 General schemes for the cross section of a tokamak with a divertor or with a magnetic separatrix. Common scenarios for the plasma-material interface. For devices with a divertor the plasma-material interaction takes place further away from the confined plasma, in a localized region on the vessel’s edge (the divertor target plates). Extra coils are responsible for producing the X-point in the poloidal magnetic field. Near the X-point are the divertor target plates, set to better withstand plasma interactions. [Adapted figure from [8]].

position of the vessel but extending along the toroidal direction corresponds to a toroidal limiter. In the second common configuration for the plasma-material interface, the divertor, extra coils are responsible for producing a null point in the poloidal magnetic field, which is also referred to as an X-point, localized in the edge as in Fig. 1.3, [8]. Near the null point are the plasma facing components for this assemble, the divertor target plates. For this configuration the plasma-material interaction takes place much further away from the plasma confinement region on the closed flux surfaces. As a consequence, is reduced the probability of contamination of the plasma core with impurities that result from the interaction between the plasma and the plasma facing components. Also reduces the probabilities on wall damage, due to the lost of particles and en-ergy outwards to the walls. Therefore a divertor is often seen in use for the majority of tokamaks.

1.4 Introduction to Plasma Physics

As previously mentioned, the positive nuclei have to overcome the Coulomb repulsion among them to be found in the short-range cross-section corresponding to the fusion reaction. The conditions of kinetic energy require very high values of temperature, for which ions are found in the tail of the Maxwellian energy distribution. Near the temperature of T ∼ 100 keV the

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cross-Figure 1.4 Lorentz force and Larmor motion. Uniform circular motion in the plane perpen-dicular to the magnetic field, with rL and wL. Helical path for charged particles with a velocity

parallel to the magnetic field. [Adapted figure from [9]].

section of the reaction on 1.1 approaches a maximum. However, such high temperatures are not required to initialize the reactions. The reference value is around T = 20 keV. Under these conditions the fuel becomes a plasma consisting of an ionized gas composed by two independent species of charged particles: the positive ions (both deuterium an tritium nuclei) and a separate specie of electrons, where a quasi-neutrality condition holds for the ion and electron densities, i.e. ni∼ ne. Also an important general characteristic is the dominance of long range electromagnetic

forces over short range collisions, and it has a major role in the collective behavior displayed by plasmas.

The Debye length, λD, in (1.4), corresponds to the characteristic decay length of charge density

and potential in a plasma, and should be smaller than the plasma size Lp, i.e. λD << Lp,

in order for the collective behavior to be dominate. In a 3-d approach of the problem (Debye shielding, [19]) the condition on (1.6) arises, [19]. The parameter in (1.6) is usually incorporated in the plasma description in order to ensure that collective behavior is dominant over collisions.

λD = ( 0Te e2_n ∞ ) 2 . (1.4) ND = n 4π 3 λ 3 D. (1.5) ND >> 1. (1.6)

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1.4.1 Guiding center drifts

Let us start by focusing on the confinement established by an imposed magnetic field, consid-ered to be constant and uniform in both time and space, B. Due to the Lorentz force, particles are forced to rotate around the the magnetic field lines with

rLe,i= v⊥me,i |Q|B (1.7) , and wLe,i = |Q|B me,i (1.8)

, where rLe,i corresponds to the Larmor radius (or gyro-radius) for electrons and ions, wLe,i to

the cyclotron frequency (or gyro-frequency), me,ito the particles mass, and Q to their respective

charges. Finally, v⊥ corresponds to the particles velocity in the direction perpendicular to the

magnetic field. Moreover, if the particles are moving in the direction parallel to the magnetic field, i.e. vk 6= 0, the result is a helical orbit as in Fig. 1.4, [9]. In a tokamak device, in

particular, the thermal velocity of the high energy particles allows them to move freely parallel to the magnetic field lines, while confined by the Lorentz force in the perpendicular direction. The gyro-motion around the field lines corresponds to the fastest motion (shortest time scale, ∝ 1/w_L_e,i) and the solutions listed below correspond to slower drifts of the center of the gyro-motion (guiding center) of each particle.

1. ~E × ~B drift

The presence of any force perpendicular to ~B, as the electric force that results from ~E⊥6= 0,

origins a drift of the guiding center in the plane perpendicular to ~B. The acceleration due to an electric force on the perpendicular plane causes rLe,i to be bigger when vLe,iincreases.

The end result is a drift in the direction perpendicular to both the electric and magnetic fields. Both ions and electrons have the same ~E × ~B velocity drift, independently of their values of charge and mass,

~ v_{E× ~}~ _B=

~ E × ~B

B2 . (1.9)

In general a force ~F⊥ perpendicular to the magnetic field results in a perturbation term

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~ v_F~_⊥× ~B= 1 q ~ F⊥× ~B B2 . (1.10) 2. ∇B drift~

If the magnetic field is not uniform in space the Larmor motion varies ∝ 1/B in such a way that the guiding center drift is according to the approximate term,

~ v∇B~ = mjv⊥2 2qB ~ B × ~∇B B2 . (1.11) 3. Curvature drift

For curved magnetic field lines a centrifugal force acts on the particles as a result of the curvilinear movement along the field direction. As mentioned before, the presence of a perpendicular force results in a drift perpendicular to both the magnetic field and the force. The curvature drift adds to the ~∇B drift and both occur in opposite directions for charges of opposing signs.

4. Diamagnetic fluid drift

The diamagnetic drift isn’t perceptible in the single particle picture. It can’t be account for single particle motion as it is not a guiding center drift. However, in the fluid picture, in the presence of a pressure gradient the fact that their are more particles moving in one direction implies a drift for the averaged velocity value of the fluid element. The perturbation from the fluid Larmor motion that results from ~∇P 6= 0, is given by

~vD = −

~ ∇P × ~B

njqjB2

. (1.12)

The ion diamagnetic drift direction is a common reference in fusion devices.

In general different values of charge, mass and temperature (kinetic energies) may lead to different velocity drifts between the plasma species, resulting in electric currents and fields.

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1.4.2 Fluid description and fluid drifts

Whether the approach is to study plasmas considering single particle motion, through ki-netic equations, or to simplify the problem describing the plasma as a Two Fluid Model, as represented below (equations (1.13) to (1.22)), a consistent result arises from every approach. Characterizing in a similar matter several particle drifts that result for charged particles in the presence of electromagnetic fields.

∂nj ∂t + ~∇(nj~vj) = 0 (1.13) Continuity Equation njmj ∂~vj ∂t + (~vj· ~∇)~vj = njqj ~ E + ~vj× ~B − ~∇Pj− ¯νjknjmj(~vj− ~vk) (1.14) Force Equation Pjn−γ_j = cte. (1.15) Energy/Equation of State Maxwell’s Equations ~ ∇ · ~B = 0 (1.16) ~ ∇ · ~E = ρ 0 (1.17) ~ ∇ × ~B = µ0J +~ 1 e2 ∂ ~E ∂t (1.18) ~ ∇ × ~E = −∂ ~B ∂t (1.19) where, ρ =X j qjnj (1.20) ~ J =X j qjnj~vj (1.21) (1.22)

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averaged macroscopic parameters such as density, velocity, temperature and pressure of the independent species composing the plasma, [19]. It provides a description of the plasma as a mixture of different fluids j, usually, found to be j = e (electorns) and i (ions) for fully ionized plasmas. However, a third specie could be considered for partially ionized plasmas still containing a considerable density of neutrals. In that case j = e, i and n. It is referred to as a Self Consistent Model, since it not only accounts for how electrons and ions respond in the presence of electric and magnetic fields, but then also to how these densities (ne, ni), and therefore the

charge and current densities riposte on the value of the fields. In this way the model accounts for the self consistent loop of dependencies occurring in the ionized plasma medium.

Each equation in the model holds for an individual species j. The first equation (1.13) is the well known Continuity equation, that corresponds to the particle conservation equation. The second equation of the model (1.14) arises from the momentum conservation for each species, commonly referred to as the Force Equation. It accounts for the momentum exchange due to the Lorentz Force acting on the particles, also due to a Pressure Force (for an isotropic medium), and finally the exchange of momentum that results from collisions between the different species in the medium. In the Force equation the last term represents the overall loss of momentum by species j when colliding with particles from species k. Seeing that νjk corresponds to the rate of

collisions between the two species, the corresponding term in1.14can be referred to as the rate of momentum loss by species j thanks to collisions with species k. The collisions between like particles are neglected since the total momentum for the species is conserved, given the overall averaged values. The third equation of the Two Fluid Model (1.15) for energy conservation is taken as an Equation of State.

The validation of the Fluid Description for a fusion plasma mainly relies on the fact that in a (toroidal) magnetic confinement device particles are trapped in the parallel direction because of the gyro-motion along the magnetic field lines, Fig. 1.2. In the perpendicular direction to the field lines this guaranties that the Nj = nj∆V particles in a fluid element ∆V act like

the particles in a highly collisional gas. The particles in a given fluid element move coherently together. Each individual particle stays confined in a physical space always within short distances relative to its neighboring particles (admitting an approximately constant Larmor radius). Each fluid element in the model is described by the macroscopic averaged parameters gj(~r, t). The

macroscopic parameters correspond to the average of all the individual particles’ values for that same property over the velocity space. In general gj(~r, t) for a fluid element centered at ~r at a

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gj(~r, t) =

R gj· fj(~r, t)dvj

R fj(~r, t)dvj

. (1.23)

Therefore loosing the need to have a detailed track of each particle’s in the velocity space.

Relative to the present work the Two Fluid Model is a good approach to further understand important velocity drifts (away from the gyro-motion along the toroidal magnetic field lines), for both electron and ion species. The velocity drifts are not only fundamental in understanding several characteristics necessary in the design of fusion devices, but also in comprehending the origin and radial convection of coherent plasma structures (called “blob-filaments”).

Taking advantage of a simplification of the Force Equation, in which we neglected the momentum transfer due to all collisions, equation (1.24) below,

njmj d~vj dt = njqjE + n~ jqj ~vj× ~B − ~∇P_j (1.24) Force Equation            d~v_Lj dt = qj mj ~vLj× ~B Larmor motion ~vL d~vE× ~~ _Bj dt = qj mj ~ E E × ~~ B drift ~v_{E× ~}~ _B d~v_Dj dt = 1 njmj − ~∇Pj Diamagnetic drift ~vD (1.25)

, it is easy to highlight some aspects on the right hand-side that are helpful in comprehending the velocity drifts. In (1.24) the second term in the right-hand side corresponds to the fluid gyro-motion ∝ d~vLj/dt. Still on the right-hand side it’s represented the velocity drift that arises

from perpendicular magnetic and electric fields (first term), the ~E × ~B drift ~v_{E× ~}~ _B. The third

term corresponds to the diamagnetic drift due to the presence of a pressure gradient ~∇P .

From the Two Fluid Model can be derived the single-fluid model MHD (magnetohydrody-namics). The MHD model is more often found to analyze the macroscopic equilibrium and stability of a fusion plasma than the previous Two Fluid Model. The MHD model can be ob-tained from the two-fluid analysis by reducing to single-fluid variables, while considering the length and time scales that characterize macroscopic plasma behavior, [13]. Such as the radius

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(a) of a plasma volume being used to define the appropriate length scale L, (L ∼ a), while the ion thermal transit time (a/vT i) across the plasma being taken as the the appropriate time scale

τ , (τ ∼ a/vT i).

As previously mentioned, there are a couple of approaches to study and describe fusion plasmas providing similar results as the ones introduced in section 1.4 for guiding center and fluid drifts. The expressions provided through 1.4are analyzed with more care in [19] and [21], accompanied with extended information on the approximations and considerations taken in their derivations.

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Chapter 2 Edge turbulence in fusion devices

2.1 Introduction

In the edge of tokamaks and general fusion devices a major challenge is faced regarding enhanced losses of particles and energy. The performance of tokamaks drastically diminishes as enhanced losses at the edge degrade the confinement conditions.

The edge plasma is generally understood as starting a few centimeters inside the last closed flux surface (LCFS) (in limited devices) or magnetic separatrix (in diverted machines), where still lays the confined plasma on closed flux surfaces. The region extends up to the scrape-off layer plasma (SOL), where opened field lines are seen to be interrupted by material structures, (chap-ter 1). In the edge plasma the neutral particle density is not negligible and therefore atomic processes strongly influence the local particle and energy balance.

It has been understood from experimental investigations, mainly over the last couple of decades, that highly localized density structures are frequently born near theLCFS. A strong theoretical hypothesis is that it is due to the nonlinear saturation of turbulence (i.e. small-scale instabilities) in the tokamak plasma boundary.

Turbulence induced structures and associated electrostatic fluctuations of plasma quantities are routinely observed in the edge and SOL plasma, this phenomenon dominates the particle and energy losses and greatly limits the overall confinement conditions. In this sense, numerous edge studies, in distinctively parameterized devices, aim to gadder a solid understanding of the physics processes in the origin of the edge turbulence and the limitations that they set on plasma confinement. An effort to also prevent the enhanced plasma interaction and damaging on plasma facing components (PFCs), contamination of the vessel with impurities, and the influence on the

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particle recycling and plasma exhaust processes. Several investigations have already determined that losses in tokamak devices are not continuous, but, instead, take place in an intermittent manner due to the radial convection of turbulent induced structures known as filaments (or blobs), [12,13,22–24].

The study of different experimental configurations in which turbulence arises can also give an insight on the instability mechanisms in its origin, therefore helping to understand how to reduce the dominant losses.

Conjointly with the time and space structure of turbulent filaments, a large body of work have been dedicated into characterizing their statistical distributions. In both subjects a wide set of results has been gathered showing a large coherency of the edge turbulence characteristics for most fusion devices. Previous results can be found from [14] to [25].

This chapter will reference a basic model for the convected particle losses by turbulence induced structures, [14]. In order to clarify the physics mechanisms in which the filament struc-tures degrade the performance of fusion plasmas. It is also presented general plasma properties that should be expected in turbulence dominant regions, setting a useful background for the results presented and discussed in chapters 4and 5.

2.2 Review on edge turbulence properties

The experimental investigations of the turbulence induced structures will provide a picture on the conditions in which these mechanisms dominate in the edge plasma. However, the mul-titude of plasma instability mechanisms driving turbulent regions is not thoroughly scooped in this study.

The induced filaments are generally characterized as magnetic-field-aligned plasma struc-tures, [14], which present considerable higher density than that of the background plasma. They are very localized in the perpendicular direction (having perpendicular scale lengths in-termediate between the ion gyro radius and macroscopic machine dimensions), while extending along the parallel direction. On [14] a working definition can be found that tries to encompass the theoretical and experimental investigations focused on the blob objects. From which it is worth to add that the plasma blob-filaments have a single-picked density distribution, with a maximum typically 2–3 higher than the surrounding rms fluctuations of the background plasma. Secondly, the filament variation along the magnetic field, to which they are highly align to, is

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Figure 2.1 Results from cameras installed in the JET tokamak. In the left is showing a 2D density plot (intensity map) in the perpendicular plane. In the perpendicular view, high density blob-like structures are shown just outside the separatrix (dashed black line). On the right, the high density structures at theJET tokamak are seen as brighter filaments, extending along the toroidal field (dashed red line). [Adapted figure from [10]].

much weaker than the variation taking place in the perpendicular direction. At last, the blob motion is dominated by a convective ~E × ~B velocity component as a result of a charge polarizing force.

2.2.1 Spatial and temporal structures

The large number of edge investigations have been a great lever in clarifying the previous spatial picture of highly localized structures in the perpendicular plane, having a blob-like two-dimensional (2D) spatial structure, while extending over the torus, in the parallel direction, with a filament three-dimensional (3D) structure. The temporal and spatial structures of the ”blob-filaments”, as named on [14], have been interpreted using various different diagnostics and analysis techniques. Langmuir Probes, which consist in electrodes inserted in the edge plasma, are the most common and simplest diagnostic used in edge/SOLstudies. Nonetheless, with the pressing need into understanding the edge turbulence processes, a large number of imaging di-agnostics such as beam emission spectroscopy (BES), gas-puff imaging (GPI), and fast cameras have been well developed, and provide an intuitive and clarifying picture of the blob-filament structures.

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plane (2D density plot) the resembling ”blob” objects of higher density over the surrounding background plasma. The example is obtained from imaging diagnostics in the Joint European Torus (JET) experiment, [10]. On the 3D image, also obtained from the JETtokamak, the 3D filament structures from the extended blobs along the parallel direction are shown.

On camera images from the Mega Amp Spherical Tokamak (MAST) experiment, Fig. 2.2, edge filaments of higher density are evidenced through brighter pixels in the images. The filament structures not only show to extend around the chamber, but also to be highly aligned with the magnetic field, having the same helical geometry along the torus. The helical geometry of the magnetic field was introduced on chapter1, Fig. 1.2.

Additional observations of 2D blob structures in the perpendicular plane are shown in Fig. 2.3 and Fig. 2.4. The former from the GPI diagnostic in the National Spherical Torus Ex-periment (NSTX tokamak), [11], an the latter obtained with the BES diagnostic on the edge of the DIII-D tokamak, [12,26]. The high density 2D blob structures, coherently seen in these measurements, generally fall between 1–3 cm (along ~r), [14] and are seen to propagate radially with typical velocities of ∼ 0.5–2 km/s.

On the other hand, theory predicts that while the higher density blobs are convected from the edge of the confined plasma through the SOL, regions of reduced density (density holes) propagate in the opposite direction (inward into the confinement region), as evidenced in the references [126, 129, 130, 144, 158, 159, 169, and 180] from [14]. To corroborate the theoretical scenario of plasma holes, simulations with seeded holes can be used to illustrate their inward motion, [27]. The formation of density holes and blobs due to plasma instabilities near the edge of the confined plasma can be read in more detail in [28,29].

On Fig. 2.4, showing the example from the BES diagnostic obtained at the DIII-D toka-mak [12,26], it’s seen not only the high density features corresponding to the blob-like structures (in red), but also low density features near the LCFS (in blue). The image shows two frames, with a time difference of 6 µs. In the figure it is seen a radial and poloidal displacement of the high density blobs, which are marked by a dashed circle in both frames.

2.2.2 Propagation mechanism by a charge polarizing force

The plasma filaments provide a mechanism for the radial outwards transport of particles and energy, resulting in losses from the edge boundary region in which they are born. A blob

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trans-Figure 2.2 (a, b) Camera Image from the MAST tokamk. Images obtained from the MAST tokamak (a,b) are processed in order to evidence the high density filament structures corre-sponding to brighter pixels (c).

The filament structures not only show to extended onward in the chamber, but also to be highly aligned with the toroidal magnetic field, having the same helical geometry along the torus. [This figure is a cortesy of Tom Farley (MAST, CCFE), 2019].

port mechanism originated by a polarizing force is summarized in [14]. The overall idea is that an external polarizing force ultimately results in an ~E × ~B velocity drift that drives the filaments.

To begin, radial or outwards expansion forces result in a poloidal ~F × ~B particle drift. This mechanism is in the origin of the vertical ∇ ~B and curvature drifts explained in chapter 1. Any force ~F in the perpendicular plane results in ~F × ~B drifts also along the perpendicular plane. In the particular cases of the ∇ ~B and the curvature drifts the displacement of ions an electrons is unequal (and have opposing signs) according to the definition on (1.10), which is dependent